Quadratic functions are the building blocks of curved relationships in math. They create U-shaped graphs called parabolas, which have unique features like a vertex and axis of symmetry. Understanding these functions helps us model real-world situations and solve optimization problems.

Graphing quadratic functions involves identifying key points and using different forms of the equation. We can find the maximum or minimum values, which are crucial for solving practical problems. Quadratic equations can be solved using various methods, each with its own strengths in different situations.

Quadratic Functions and Parabolas

Features of parabolas

• Parabola: U-shaped curve symmetric about the axis of symmetry
  • Vertex: Point where the parabola changes direction, either a minimum or maximum point
    • If $a > 0$ in $f(x) = ax^2 + bx + c$, the parabola opens upward with a minimum at the vertex (cup-shaped)
    • If $a < 0$, the parabola opens downward with a maximum at the vertex (cap-shaped)
  • Axis of symmetry: Vertical line passing through the vertex, dividing the parabola into two equal halves
    • Equation of the axis of symmetry: $x = -\frac{b}{2a}$, where $a$ and $b$ are coefficients in $f(x) = ax^2 + bx + c$
  • Other key features include the y-intercept (point where the parabola crosses the y-axis) and x-intercepts or roots (points where the parabola crosses the x-axis)
• Domain and range: The domain of a quadratic function is all real numbers, while the range depends on the direction of opening and the vertex

Graphing quadratic functions

• Standard form of a quadratic function: $f(x) = ax^2 + bx + c$
  • $a$: Determines the direction of opening and width of the parabola
    • If $|a| > 1$, the parabola is narrower than the parent function $f(x) = x^2$
    • If $0 < |a| < 1$, the parabola is wider than the parent function
  • $b$: Affects the location of the axis of symmetry and vertex
    • The sign of $b$ determines the horizontal shift direction of the vertex from the origin
  • $c$: Vertical shift of the parabola from the origin
• Vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$
  • $(h, k)$ is the vertex of the parabola
  • To graph, locate the vertex $(h, k)$, then plot points on either side using $a$ to determine width and direction of opening
• Graphing steps:

1. Identify the key features (vertex, axis of symmetry, y-intercept, x-intercepts)
2. Plot the vertex and y-intercept
3. Use the axis of symmetry to plot symmetric points on either side of the vertex
4. Connect the points to form the parabola

Extrema of quadratic functions

• The minimum or maximum value of a quadratic function occurs at the vertex
  • For $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is $x = -\frac{b}{2a}$
  • Substitute the x-coordinate into the original function to find the y-coordinate of the vertex
• If the parabola opens upward ($a > 0$), the vertex is a minimum point (lowest point on the graph)
• If the parabola opens downward ($a < 0$), the vertex is a maximum point (highest point on the graph)
• The extrema can be used to solve problems involving optimization, such as finding the maximum profit or minimum cost

Applications of quadratic optimization

• Optimization problems involve finding the maximum or minimum value of a quadratic function in a real-world context
  • Identify the given information and the quantity to be optimized (area, profit, distance)
  • Define variables and express the quantity as a quadratic function
  • Find the vertex of the quadratic function to determine the maximum or minimum value
  • Interpret the result in the context of the problem
• Example: A farmer has 100 m of fencing to enclose a rectangular garden. What dimensions will result in the maximum garden area?

1. Let $x$ be the width and $y$ be the length of the garden

2. Perimeter: $2x + 2y = 100$, so $y = 50 - x$

3. Area: $A(x) = xy = x(50 - x) = 50x - x^2$

4. Find the vertex of $A(x)$ to determine the maximum area and corresponding dimensions

   • Vertex: $x = -\frac{b}{2a} = -\frac{50}{2(-1)} = 25$, so the width is 25 m and length is $50 - 25 = 25$ m
   • Maximum area: $A(25) = 50(25) - 25^2 = 625$ m²

Solving Quadratic Equations

• Factoring: A method to find the roots of a quadratic equation by expressing it as a product of linear factors
• Completing the square: A technique used to rewrite a quadratic equation in vertex form, which can be used to solve the equation or find the vertex
• Quadratic formula: A general formula for solving quadratic equations in the form $ax^2 + bx + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Discriminant: The expression under the square root in the quadratic formula ($b^2 - 4ac$), which determines the nature of the roots of a quadratic equation